The need to reduce the length of queue (waiting time) forms the basis of this research. This project work centers on the queuing system witnessed at the Nekede bus terminal; and a single serve queuing system was adopted in the analysis. The basic aim and objectives of this research is to identify the distribution of the arrival and service and finding out if increasing the number services (terminal) would tend to reduce the waiting time in the system. Different probability distribution where used to analyze the data which lead to the computing of the queuing statistics with busy time to be 0.73 and idle time to be 0.27, number of vehicle in the system to be 3, number vehicle in queue to be 2, expected waiting time in system to be 0.23 mins and expected waiting time in queue to be 0.17 mins. And also the total number of vehicle served to be 190 within 695 minutes. In conclusion, the researcher finds out that the arrival of buses follows a Poisson distribution as well as exponential service times. And also since there is almost 0 queue (i.e. ρ<1), there is no need for more servers in the system.

CHAPTER ONE

1.0 INTRODUCTION

Waiting in lines is a part of our everyday life. Waiting in lines may be due to the demand at any one time may be more than the capacity of service (over crowded), overfilling or due to congestion. Any time there is more customer demand for a service than can be provided, a waiting lines forms. We wait in lines at the movie theatre, at the bank for a teller, at hospital (s) for diagnosis and treatment at a grocery store. Wait time is depends on the number of people waiting before you, the number of servers serving line, and the amount of service time for each individual customer. Customers can be humans or an object such as customer orders to be process, a machine waiting for repair. Matte analytical method of analyzing the relationship between congestion and delay caused by it can be modelled using queuing analysis. Queuing theory provides tools needed for analysis of system of congestion. Mathematically, systems of congestion appear in many diverse and complicated ways and can vary in extent and complexity.

A waiting line system or queuing system is defined by two important elements: the population source of its customers and the process or service system. The customer population can be considered as finite or infinite. The customer population is finite when the number of customers affects the potential new customers for the service system already in the system. When the number of customers waiting in line does not significantly affect the rate at which the population generates new customers, the customer population is considered infinite. Customer behaviour can change and depends on waiting line characteristics. In addition to waiting, a customer can choose other alternative. When a customer enters the waiting line but leaves before being serviced, process is called Reneging. When customers changes one line to another to reduce wait time, process is called jockeying. Balking occurs when a customer do not enter waiting line but decides to come back later. Another element of queuing system is service system. The number of waiting lines, the number of services, the arrangement of the services, the arrival and service patterns, and the service priority rules characterized the service system. Queue system can have channels or multiple waiting lines. Examples of single waiting line are bank counter, airline counters, restaurants, amusement parks etc. In these examples multiple servers might serve customer. In the single line multiple servers has better performance in terms of waiting times and eliminates jockeying behaviour than the system with a singe line for each server. System serving capacity is a function of the number of service facilities and severs proficiency. In queuing system, the terms server and channel are used interchangeable. Queuing systems are either single server or multiple servers. Single server examples includes gas station food mart with single checkout counter, a theatre with a single person selling ticket and controlling admission into the show. Multiple server examples include gas station with multiple gas pumps, grocery stores with multiple cashiers, and multiple tellers in a bank. Services require single activity or services of activities called phases. In a Single-phase system, the service is completed all at once, such as a bank transaction or grocery store checkout counter. In a multiple phase system, the service is completed in a series of phases, such as at fast food restaurant with ordering, pay, and pick up windows. Queuing system is characterised by rate at which customers arrive and served by service system. Arrival rate specifies the average number of customers per time period. The service rate specifies the average number of customer that can be served during a time period.

The service rate governs capacity of the service system.

It is the fluctuation in arrival and service patterns that causes wait in queuing system. Waiting line models assume that customers arrive according to a Poisson probability distribution, and service times are described by an exponential distribution. The poisson distribution specifics the probability that a certain number of customers will arrive in a given time period. The exponential distribution described the service times as the probability that a particular service time will be less than or equal to a given amount of time. A waiting line priority rule determines which customer is served next. A frequently used priority rule is first come first served. Other rules include best customers first, high-test profit customer first, emergencies first, relatives and friends first, closest units first, last in first served, quickest to service first, Elderly people, women and children first, random order, very important person (VIPS) first and so on. Although each priority rule has merit, it is important t use the priority rule that best supports the overall organization strategy. The priority rule used affects the performance of waiting line system.

ASSUMPTION INHERENT IN QUEUING SYSTEM

Basic single server model assumes customers are arriving at poisson arrival rate with exponential service times and first come, first services queue discipline, and infinite queue length, and infinite calling population. By adding additional resources to single server system either service rate can be decreased with additional cost overhead. In single server single-phase system, customer is served once completed.

Common examples of single server singe-phase are teller counter in a bank, a cashier counter in super, market, automated ticketing machine at rain station. In single server queuing system wait time or performance of system depend on efficiency of serving person or service machine. Single server single-phase queuing system is most commonly automated system found in our regular life. For examples many superstores have replaced manual counters with automated machines. Single server multiple-phase incorporates division of work into phase to keep waiting line moving as completion of whole complete operation might increase wait in a line. Common examples of the these systems are automatic or manual car wash drive through restaurants.

QUEUING THEORY

Queuing Theory is the mathematical study of waiting lines (or queues).

The theory enables mathematical analysis of several related processes including arriving at the (back of the) queue, waiting in the queue (essentially a storage process), and being served by the servers at the front of the queue. Theory permits the derivation and calculation of several performance measures including the average waiting time in the queue or the system, the expected number waiting or receiving and the probability of encountering the system in certain states, such as empty full, having an available server or having to wait a certain time to be served.

Queuing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide service. It is applicable in a wide variety of situations that they may be encountered in business, commerce, industry, public service and engineering. Applications are frequently port and telecommunication. Queuing theory is directly applicable to intelligent transportation systems, call centers, and traffic flow.

ELEMENTS OF A QUEUING MODEL

The principal actors in a queuing situation are the customer and the server. Customers are generated from a source. On arrival at a service facility, they can start service immediately or wait in a queue if the facility is busy. When a facility completes a service, it automatically “pulls” a waiting customer, if any, from the queue. If the queue is empty, the facility becomes idle until a new customer arrives. customers, and the service is described by the service time per customer.

1.1 STATEMENT OF PROBLEM

Waiting for service is part of our daily life. We wait to eat in restaurants, we “queue up” at the check-out counters in grocery stores, and we “line up” for service in post offices. And the waiting phenomenon is not an experience limited to human beings only: Jobs wait to be processed on a machine, planes circle in a stack before given permission to land at an airport, and cars can stop at traffic lights. Waiting can not be eliminated completely without incurring inordinate expenses and the goal is reduce its adverse impact to “tolerable” levels. Basic single server model assumes customers are arriving at Poisson arrival rate with exponential service times.

It therefore becomes an interest for the researcher to find out what can be done to reduce the queue and to identify the distribution which the arrival and service time follows using Nekede bus terminal as a case study.

1.2 AIMS AND OBJECTIVES OF THE STUDY

The major aims and objective of this study are as follows:

1. To find out if the arrival of buses in a terminal follow Poisson distribution

2. To find out if the service times are exponential.

3. To find out if increasing the number of servers (terminals) would reduce the queue/waiting time

4. To identify if the queue operates in a steady state condition.

5. To identify, show, then suggest through empirically backed decision how the management of Nekede bus terminal go about the reduction of waiting time of vehicles in the terminal.

RESEARCH QUESTIONS

1. What are the causes of queuing in the terminal?

2. Does the arrival of buses in the terminal follow Poisson distribution?

3. Do their service times follow exponential?

1.3 STATEMENT OF HYPOTHESIS

Ho: The arrival of buses follows Poisson distribution

H1:The arrival of buses does not follow Poisson

Distribution

Ho: The service times follow exponential distribution

H1:The service times does not follow exponential

Distribution.

1.4 SIGNIFICANCE OF STUDY

1. This study will be immense benefit to public and private sector to determine when to open another branch (terminal).

2. It will be of immense benefit/guide to the management of Nekede bus terminal and other bus terminal management to know the strategy/system to adopt in reduction of waiting lines.

1.5 SCOPE OF STUDY

Queuing system have a wide area of application in every field of human endeavours. For purpose of this project research work, the study is limited to Nekede bus terminal.

1.6 DEFINITION OF TERMS

Queues: can be define as items waiting for service

Queuing: is the study of queues.

Queue Discipline: represent the order in which customer

are selected from queue for service.

Jockeying: Is when a customer leaves one queue to

another is hope of reducing waiting time.

Reneging: When the customer leaves the queue because

He/she has waited too long.

Balking: Occurs when customer do not enter waiting line but decides to come back later.

N = Number of customers in the system (in queue plus

in service

l_n= Arrival rate given n customers in the system.

m_n= Departure rate given n customers in the system

P_n = Steady state probability of n customers in the

System

L_s= Expected number of customers in system. (Length

of service)

L_q = Expected number of customers in queue. (Length of

queue)

W_s = Expected waiting time in the system

Of queue)

W_q = Expected waiting time in queue

C = Expected number of busy servers

Traffic intensity: This is the proportion of the average arrival rate to the average service rate. It indicates the extent and propensity of queuing. It is calculated thus:

Traffic intensity = mean arrival rate per unit time

Mean service rate per unit time

A simple mathematical presentation for the traffic intensity